Methods2024 NHT Exam 2 Section B · Q5CAS active12 marks

Exponential functions and their inverses

Let $f: \R \to \R, f(x) = e^{-ex}$ .

Show that the rule and domain for $f^{-1}$ , the inverse function of $f$ , is

$f^{-1}(x) = -\frac{1}{e}\log_e(x)$

where $x \in (0, \infty)$

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

The graphs of $f$ and $f^{-1}$ are shown below. Their unique intersection point $\left(\frac{1}{e}, \frac{1}{e}\right)$ is also labelled.

Find the exact area of the shaded region that is bounded by the graphs of $f$ , $f^{-1}$ and the coordinate axes.

$Graphs of the decreasing curve f and its inverse f^{-1} (dashed) meeting at the labelled point (1/e, 1/e). f has y-intercept 1 and f^{-1} has x-intercept 1; the region bounded by both graphs and the coordinate axes is shaded.$

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

c.i.

Let $g: \R \to \R, g(x) = e^{ax}$ , where $a \in \R \setminus \{0\}$ .

Consider the function $g_1(x) = e^{ax}$ , where $a = \frac{1}{e}$ .

Find the equation of the tangent line to the graph of $y = g_1(x)$ at $x = e$ .

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

Methods2024 NHT Exam 2 Section B · Q5CAS active12 marks

Exponential functions and their inverses

Let $f: \R \to \R, f(x) = e^{-ex}$ .

Show that the rule and domain for $f^{-1}$ , the inverse function of $f$ , is

$f^{-1}(x) = -\frac{1}{e}\log_e(x)$

where $x \in (0, \infty)$

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

The graphs of $f$ and $f^{-1}$ are shown below. Their unique intersection point $\left(\frac{1}{e}, \frac{1}{e}\right)$ is also labelled.

Find the exact area of the shaded region that is bounded by the graphs of $f$ , $f^{-1}$ and the coordinate axes.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

c.i.

Let $g: \R \to \R, g(x) = e^{ax}$ , where $a \in \R \setminus \{0\}$ .

Consider the function $g_1(x) = e^{ax}$ , where $a = \frac{1}{e}$ .

Find the equation of the tangent line to the graph of $y = g_1(x)$ at $x = e$ .

1 mark

Hints, answer checking and worked solutions come with a Season Pass.